Theorem falseral0 4081

Description: A false statement can only be true for elements of an empty set. (Contributed by AV, 30-Oct-2020.)

Assertion

Ref	Expression
falseral0	|- ( ( A. x -. ph /\ A. x e. A ph ) -> A = (/) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem falseral0

Step	Hyp	Ref	Expression
1		df-ral 2917	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
2		19.26 1798	. . 3 |- ( A. x ( -. ph /\ ( x e. A -> ph ) ) <-> ( A. x -. ph /\ A. x ( x e. A -> ph ) ) )
3		con3 149	. . . . . . 7 |- ( ( x e. A -> ph ) -> ( -. ph -> -. x e. A ) )
4	3	impcom 446	. . . . . 6 |- ( ( -. ph /\ ( x e. A -> ph ) ) -> -. x e. A )
5	4	alimi 1739	. . . . 5 |- ( A. x ( -. ph /\ ( x e. A -> ph ) ) -> A. x -. x e. A )
6		alnex 1706	. . . . 5 |- ( A. x -. x e. A <-> -. E. x x e. A )
7	5, 6	sylib 208	. . . 4 |- ( A. x ( -. ph /\ ( x e. A -> ph ) ) -> -. E. x x e. A )
8		notnotb 304	. . . . 5 |- ( A = (/) <-> -. -. A = (/) )
9		neq0 3930	. . . . 5 |- ( -. A = (/) <-> E. x x e. A )
10	8, 9	xchbinx 324	. . . 4 |- ( A = (/) <-> -. E. x x e. A )
11	7, 10	sylibr 224	. . 3 |- ( A. x ( -. ph /\ ( x e. A -> ph ) ) -> A = (/) )
12	2, 11	sylbir 225	. 2 |- ( ( A. x -. ph /\ A. x ( x e. A -> ph ) ) -> A = (/) )
13	1, 12	sylan2b 492	1 |- ( ( A. x -. ph /\ A. x e. A ph ) -> A = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  uvtxa01vtx0  26297